FAITH'S PROBLEM ON R-PROJECTIVITY IS UNDECIDABLE

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10401396" target="_blank" >RIV/00216208:11320/19:10401396 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=FynXWlgi0b" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=FynXWlgi0b</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/proc/14209" target="_blank" >10.1090/proc/14209</a>

Jazyk výsledku

angličtina

Název v původním jazyce

FAITH'S PROBLEM ON R-PROJECTIVITY IS UNDECIDABLE

Popis výsledku v původním jazyce

In Faith [Grundlehren der Mathematischen Wissenschaften. 191 (1976)], Faith asked for what rings R does the Dual Baer Criterion hold in Mod-R, that is, when does R-projectivity imply projectivity for all right R-modules? Such rings R were called right testing. Sandomierski proved that all right perfect rings are right testing. Puninski et al. [J. Algeb. 484 (2017) pp. 198-206] have recently shown for a number of nonright perfect rings that they are not right testing, and noticed that [Trans. Amer. Math. Soc. 348 (1996) pp. 1521-1554] proved consistency with ZFC of the statement &apos;each right testing ring is right perfect&apos; (the proof used Shelah&apos;s uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith&apos;s question above is undecidable in ZFC. We also provide examples of nonright perfect rings such that the Dual Baer Criterion holds for all small modules (where small means countably generated, or &lt;= 2(N0)-presented of projective dimension &lt;= 1).

Název v anglickém jazyce

FAITH'S PROBLEM ON R-PROJECTIVITY IS UNDECIDABLE

Popis výsledku anglicky

In Faith [Grundlehren der Mathematischen Wissenschaften. 191 (1976)], Faith asked for what rings R does the Dual Baer Criterion hold in Mod-R, that is, when does R-projectivity imply projectivity for all right R-modules? Such rings R were called right testing. Sandomierski proved that all right perfect rings are right testing. Puninski et al. [J. Algeb. 484 (2017) pp. 198-206] have recently shown for a number of nonright perfect rings that they are not right testing, and noticed that [Trans. Amer. Math. Soc. 348 (1996) pp. 1521-1554] proved consistency with ZFC of the statement &apos;each right testing ring is right perfect&apos; (the proof used Shelah&apos;s uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith&apos;s question above is undecidable in ZFC. We also provide examples of nonright perfect rings such that the Dual Baer Criterion holds for all small modules (where small means countably generated, or &lt;= 2(N0)-presented of projective dimension &lt;= 1).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA17-23112S" target="_blank" >GA17-23112S: Strukturní teorie reprezentací algeber (lokalizace a vychylující teorie)</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Proceedings of the American Mathematical Society

ISSN

0002-9939

e-ISSN

—

Svazek periodika

147

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

8

Strana od-do

497-504

Kód UT WoS článku

000454742000008

EID výsledku v databázi Scopus

2-s2.0-85061568935